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# Determine functions u and v satisfying given conditions

The graph of a solution of the differential equation intersects the graph of a solution of the differential equation at the origin. Determine formulas for the functions and if the curves have equal slopes at the origin and if First, we compute the general solutions of the two second-order differential equations. is of the form This gives us so and . Hence, the general solutions are The other equation is of the form Therefore, so and the general solutions are given by Letting and denote the particular solutions we are looking for (and renaming the constants and in the equations for and ) we have We are given and . So, and Finally, we are also given Putting these all together and solving for the constants we obtain Therefore, the functions we are looking for are ### One comment

1. Alex Willis says:

Did you use matrices to solve for the constants? I’m not sure how to solve for them. Thanks.